224 THE THEORY OF SCREWS. [214, 



cylindroid. In like manner let p be the other principal screw of the cylin- 

 droid (0i/r). Then p and a determine the cylindroid (per) which belongs to 

 the system, 6 must lie on the common perpendicular to p and cr, and hence 

 the screws of the cylindroid (per) each intersect 6 at right angles. 



If is a screw of stationary pitch in a four-system, it can be shown 

 that three screws p, cr, r not on the same cylindroid can be found in the same 

 system, and such that they intersect 6 at right angles. In this case p, cr, r 

 will determine a three-system, every screw of which intersects 6 at right 

 angles. 



215. Application to the Two-System. 



The principles of the last article afford a simple proof of many funda 

 mental propositions in the theory. We take as the first illustration the well- 

 known fact ( 76) that if the co-ordinates of a screw satisfy four linear 

 equations then the locus of that screw is a cylindroid. 



From the general theorem we see by the case of n = 2 that in any two- 

 system a screw of stationary pitch will be intersected at right angles by 

 another screw of the two-system. 



These two screws may be conveniently taken as the first and third of the 

 canonical co-reciprocal system lying on the axes of x and y. Hence we have 

 as the co-ordinates of a screw of the system l , 0, # 3 , 0, 0, 0. 



The investigation has thus assumed a very simple form inasmuch as the 

 four linear equations express that of the six co-ordinates of a screw of the 

 system four are actually zero. 



Let X, fjb, v be the direction angles of the screw 6 with respect to the 

 associated Cartesian axes then ( 44), 



a _ (Pe + a ) cos ^ ~ d ei sin X (p e a) cos X d ei sin X 



V\^ ) v) = I 



a a 



a (Pe + &) cos P - ^02 sin /u. (p e - b) cos p-d^ sin p 



e * = IT &quot; ; ~TF~ - ; 



_ _ ( p e + c) cos v d es sin v _ _ (p 6 c) cos v - d es sin v 

 c c 



The two last of these equations give 



cos v = ; d e3 = 0. 



Hence we learn that 6 must intersect the axis of z at right angles. 6 is 

 thus parallel to the plane of xy at a distance d ei = d ez = z, and accordingly 

 we have the equations 



( p g a) cos X z sin X = 0, 



(p 6 b) cos /j. z sin /u, = 0, 



